Random polymers on the complete graph
Francis Comets, Gregorio R. Moreno Flores, Alejandro F. Ramirez
aa r X i v : . [ m a t h . P R ] J a n RANDOM POLYMERS ON THE COMPLETE GRAPH
FRANCIS COMETS , , GREGORIO MORENO , , AND ALEJANDRO F. RAM´IREZ , , Abstract.
Consider directed polymers in a random environment on the complete graph ofsize N . This model can be formulated as a product of i.i.d. N × N random matrices andits large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure.We detail this correspondence, derive the long-time limit of the model and obtain a co-variantdistribution for the polymer path.Next, we observe that the model becomes exactly solvable when the disorder variables arelocated on edges of the complete graph and follow a totally asymmetric stable law of index α ∈ (0 , N asymptotics can be taken in this setting, for instance, for the free energy of thesystem and for the invariant law of the polymer height with a shift. Moreover, we give someperturbative results for environments which are close to the totally asymmetric stable laws. Introduction and results
Directed polymers in random environments were introduced in [27] as a model for the phaseseparation line of the 2 d Ising model in the presence of impurities. Since then, they have beenthe subject of an important body of work both in the mathematics and physics communities.On the mathematics side, their study started with the work [30], shortly followed by [7]. Anup-to-date account on the mathematical treatment of directed polymers can be found in [15].On the physics side, this model was often studied in connection with growing surfaces. Inparticular, it was observed that the Kardar-Parisi-Zhang equation [32] can be considered as acontinuum version of directed polymers [33].In its usual discrete version, the model can be formulated in terms of two basic ingredients: (i)The polymer paths are given a-priori by the trajectories of a simple symmetric random walkon Z d starting at the origin, whose law we denote by P ; (ii) The environment is given by afamily { η ( t, x ) : t ≥ , x ∈ Z d } of i.i.d. random variables with common distribution P . Then,given t ≥ β > x of length t as the probability measure µ ηt,β ( d x ) = 1 Z ηt,β exp { β t X s =1 η ( s, x s ) } P ( d x ) , where Z ηt,β is the normalizing constant. Date : November 23, 2017.AMS 2000 subject classifications . Primary 60K35; secondary 82B23, 60B20 .
Key words and phrases.
Directed polymers, random medium, exactly solvable model, stable laws, product ofrandom matrices. Universit´e Paris Diderot. Partially supported by CNRS, Laboratoire de Probabilit´es et Mod`eles Al´eatoires,UMR 7599. Pontificia Universidad Cat´olica de Chile. Partially supported by Fondecyt grant 1171257. Pontificia Universidad Cat´olica de Chile. Partially supported by Fondecyt grant 1141094. Partially supported by N´ucleo Milenio ‘Modelos Estoc´asticos de Sistemas Complejos y Desordenados’. Partially supported by Mathamsud ‘Large-scale Behavior of Stochastic Systems’.
In this note, we propose to replace, in the above definition, Z d with its natural graph structureby the complete graph with N sites. The precise definition of the model is deferred Section 1.1.Alternatively, this can be seen as a polymer with long-range jumps on a finite state space.Large finite state spaces have been considered in the literature as approximations of infinitesystems, for instance [10] for directed polymers on a cylinder, or directed polymers on n -treesas in [18]. In [11], a zero-temperature version of our model was introduced to compute thecorrections for large finite systems to continuous limit equations of front propagation.There are several additional reasons to consider polymers on other graphs. First of all, theyoffer several simplifications compared to the original model on Z d or R d . In this respect,models on the complete graph can be seen as a mean field approximation, much in the sameway the Curie-Weiss model relates to the Ising model. When considering mean-field versionsof directed polymers it is more common to refer to polymers on the tree [12] [20], but thesemodels show fundamental differences with the original ones [15]. Our model on the completegraph preserves some features, and can also be seen as a positive temperature version of thelast passage percolation studied in [11, 16] from where we draw many ideas.This approximation of infinite graphs by finite ones can be made quantitative. For instance,the paper [22] considers a particular product of random symplectic matrices corresponding toa random diffusion on the d -dimensional discrete torus. Assuming d ≥ apriori law on the path space is an ergodic Markov chain.Finally, on the complete graph, we are able to find a law on the environment that makes themodel solvable in the sense that the law of the properly normalized partition function canbe computed explicitly. This is analogous to the zero-temperature version of the model from[11, 16, 19]. Exactly solvable models are rare in statistical mechanics, but they yield mostinformative results. This one seems to be new. In the case of polymers on the lattice, we canhowever cite [39] for the original model on Z and [36] for a semi-continuous counterpart.The study of polymer models on finite graphs falls in the scope of products of random matrices.More precisely, the free energy is related to the top Lyapunov exponent of such products forwhich a complete theory has been established in the literature initiated in [25] and developpedin particular in [8, 14, 24, 26, 28, 29, 35]. This formalism allows us to derive, for example, theexistence of the free energy as well as Gaussian fluctuations for the logarithm of the partitionfunction. In different direction, we mention [1] which gives a nice account on Perron-Frobeniustheory for product of random matrices together with its relation to thermodynamic formalism.For ”infinite matrices” – i.e. for random positive operators in infinite dimension, limits mayfail to exist due to lack of compactness. The authors in [2, 37] deal with products of randomoperators on the whole Z d , but some localization features – a priori embedded in the specialmodels – make them essentially compact.1.1. Directed polymers on the complete graph.
We study the following model of directedpolymers on the complete graph with N sites: for t ≥ ≤ i, j ≤ N , consider theset of paths starting at location i at time 0 and ending at j at time t , J N (0 , i ; t, j ) = (cid:8) j = ( j , · · · , j t ) : 1 ≤ j s ≤ N, ∀ ≤ s ≤ t − ; j = i, j t = j (cid:9) . (1.1)Let { ω i,j ( t ) : 1 ≤ i, j ≤ N, t ≥ } be a family of i.i.d. positive random variables defined onsome probability space (Ω , A , P ). For a fixed realization of the environment and t ≥
1, we
ANDOM POLYMERS ON THE COMPLETE GRAPH 3 define the point-to-point (P2P) polymer partition function Z N (0 , i ; t, j ) = X j ∈ J N (0 ,i ; t,j ) t Y s =1 ω j s − ,j s ( s ) . (1.2)Viewed as a function of j , its logarithm is referred to as the polymer height function . We denotewith a ⋆ quantities Q ( ⋆ ) which are, depending on the context, unions or sums over locations j ∈ { , . . . N } of Q ( j ). For instance, J N (0 , i ; t, ⋆ ) = ∪ Nj =1 J N (0 , i ; t, j ), and Z N (0 , i ; t, ⋆ ) = P Nj =1 Z N (0 , i ; t, j ) is the so-called point-to-line (P2L) polymer partition function. Similarly, J N (0 , ⋆ ; t, j ) = ∪ Ni =1 J N (0 , i ; t, j ), and Z N (0 , ⋆ ; t, j ) = P Ni =1 Z N (0 , i ; t, j ) is the line-to-point (L2P)partition function.Our analysis will rely on a tight relation between our model and products of random matriceswhich emerges from the following observation: defining the N × N matrix Π( t ) as the productΠ( t ) = X (1) X (2) · · · X ( t ) , of the matrices X ( t ) = [ ω i,j ( t )] i,j , we see that the P2P partition function (1.2) is the ( i, j )-entryof Π( t ), Z N (0 , i ; t, j ) = Π( t ) i,j . (1.3)(We do not indicate in the notation the dependence in N of X and Π, although we will let N go to infinity at some stage.) Let us denote by Z N ( t ) the (column) vector given by the L2Ppartition functions Z N ( t ) = (cid:0) Z N (0 , ⋆ ; t, , · · · , Z N (0 , ⋆ ; t, N ) (cid:1) ∗ where M ∗ denotes the transposed of the matrix M . Our convention in the paper is that allvectors are column vectors, and we write v ( j ) for the j -th coordinate of v , so that Z N ( t, j ) ≡Z N (0 , ⋆ ; t, j ). From (1.3), we obtain Z N ( t ) ∗ = ∗ Π( t ) , (1.4)where ∗ denotes the N -dimensional row vector with all entries equal to 1. Similarly, the vectorof P2L partition functions can be written as Π( t ) .This point of view allows us, among other things, to relate the free energy of the model to theLyapunov exponent of products of i.i.d. random matrices. In our case the matrices have anadditional feature – entries are i.i.d. – but the theory applies to the general case under mildassumptions. At this point, it is convenient to take (1.4) as the starting point of our analysisand consider the slightly more general framework of the recursion Z N ( t ) ∗ = Z N ( t − ∗ X ( t ) (1.5)allowing general initial conditions Z N (0) ∈ R N + \ { } ..We follow the formalism of [28, 29] based on the action of products of random matrices onprojective spaces. For v ∈ R N + \ { } and α >
0, define the α -norm of v as || v || α = ( P Nj =1 v αi ) /α .Of course, this quantity is a norm only in the case α ≥
1. Next, we introduce the α -symplex¯ B α = { v ∈ R N + : || v || α = 1 } , together with the projection Ψ α ( v ) = v || v || α from R N + \ { } onto ¯ B α .For v ∈ R N + \ { } and X an N -by- N matrix with positive entries, we define the product α · by X α · v := X v || X v || α ∈ ¯ B α . We will drop the subscripts and superscripts from the notation when α = 1 and write¯ B := ¯ B , X · v := X · v. ANDOM POLYMERS ON THE COMPLETE GRAPH 4
Finally, define X N,α ( t ) := Ψ α ( Z N ( t )) = Z N ( t ) || Z N ( t ) || α ∈ ¯ B α , (1.6)and, again, write X N := X N, . This leads to the simple decompositionlog Z N ( t, i ) = log || Z N ( t ) || α + log X N,α ( t, i ) (1.7)Note that, by the recursion (1.5) and homogeneity, we have X N,α ( t ) = Ψ α (cid:0) X ( t ) ∗ Z N,α ( t − (cid:1) = Ψ α (cid:0) X ( t ) ∗ X N,α ( t − (cid:1) , showing that { X N,α ( t ) : t ≥ } is a Markov chain. We list further important properties of thischain in the next theorem.1.2. Product of random matrices and polymer model structure.
All three results inthis subsection come as applications of the general theory of product of independent randommatrices. They are overlooked in this context, although they provide a complete understandingof the model for a fixed N . For integers s < t , letΠ( s, t ) = X ( s + 1) . . . X ( t ) , Π( t, t ) = I N , Π( t ) = Π(0 , t ) . Theorem 1.1.
Let Z N (0) ∈ R N + . (1) For all α > , the recursion (1.5) with initial condition Z N (0) defines by (1.6) a time-homogeneous Markov chain { X N,α ( t ) : t ≥ } with values in ¯ B α . (2) There exists an event Ω with P (Ω ) = 1 such that the (random) limit V ∞ N,α = lim t →∞ Π( t ) α · v, (1.8) exists for all α > , ω ∈ Ω and does not depend on v ∈ R N + . Moreover, V ∞ N,α = Ψ α ( V ∞ N,β ) for all α, β > . (3) Let m N,α denote the law of V ∞ N,α . The chain ( X N,α ( t )) t ≥ with initial law m N,α is sta-tionary and ergodic. (4)
Denote by θ s the shift on Ω by s ∈ Z , θ s ω ( t ) = ω ( s + t ) , and set V ∞ N,α ( s ) := V ∞ N,α ◦ θ s = lim t →∞ Π( s, t ) α · v, (1.9) and V ∞ N,α ( s, j ) the j -th component of this vector. (In particular, V ∞ N,α (0) = V ∞ N,α .) Then, X (0) α · V ∞ N,α = V ∞ N,α ( −
1) (1.10)This is proved in Section 2.1 and the Appendix.Our next result states the almost sure existence of the free energy as well as Gaussian fluc-tuations for the logarithm of the partition function. From (1.7), note that the asymptotics oflog Z N ( t, i ) is essentially given by that of the first term. That this limit is independent of α comes from the observation that, for each α >
0, there exists a constant c N ( α ) ∈ (1 , ∞ ) suchthat c N ( α ) − || v || ≤ || v || α ≤ c N ( α ) || v || , ∀ v ∈ R N + . Theorem 1.2.
Fix N , assume that the ω ’s are not constant and that E | log ω i,j | δ < ∞ forsome positive δ . Then, there exist numbers v N and σ N > such that, for all j = 1 , . . . N , lim t →∞ t log Z N ( t, j ) = v N a . s ., and √ t (cid:0) log Z N ( t, j ) − v N t (cid:1) law −→ N (0 , σ N ) as t → ∞ . Furthermore, with V ∞ N,α from (1.8) , v N = E (cid:2) log || X (0) V ∞ N,α || α (cid:3) . (1.11) ANDOM POLYMERS ON THE COMPLETE GRAPH 5
We give a proof in Section 2.2.We now turn to the asymptotic of the polymer measure. The P2L polymer measure starting at i with time-horizon T , denoted by P ω ,i ; T,⋆ , is the (random) probability measure on J N (0 , i ; T, ⋆ )given by P ω ,i ; T,⋆ (cid:0) j = ( j , · · · , j T ) (cid:1) = 1 Z N (0 , i ; T, ⋆ ) T Y s =1 ω j s − ,j s ( s ) . (1.12)Similarly to Theorem 1.1, there exists an almost-sure limit to the “backwards-in-time” product ←− V ∞ N,α ( s ) = lim t →−∞ Π( t, s ) ∗ α · v (1.13)which does not depend on v ∈ R N + . Since X (0) ∗ law = X (0), we have ←− V ∞ N,α ( s ) law = V ∞ N,α . (1.14)Our second set of results states the existence of an infinite volume polymer measure togetherwith a co-variant law: define the random probability measure ν N ( t, · ) on { , . . . N } by ν N ( t, j ) := ←− V ∞ N,α ( t, j ) V ∞ N,α ( t, j ) P Nk =1 ←− V ∞ N,α ( t, k ) V ∞ N,α ( t, k ) = ←− V ∞ N ( t, j ) V ∞ N ( t, j ) P Nk =1 ←− V ∞ N ( t, k ) V ∞ N ( t, k ) , (1.15)since the ratio in the second term does not depend on α , by Theorem 1.1, point 2. In words,the co-variant law is proportional to the doubly infinite product of weights over polymers (fromtimes −∞ to + ∞ ) which take the value j at time t . Theorem 1.3. (1)
For almost every environment ω , the polymer measure P ω ,i ; T,⋆ convergesas T → ∞ to the (time-inhomogeneous) Markov chain with P ω ( j = i ) = 1 and transi-tion probabilities given by P ω ( j t +1 = ℓ (cid:12)(cid:12) j t = k ) = ω k,ℓ ( t +1) V ∞ N ( t +1 , ℓ ) P Nℓ ′ =1 ω k,ℓ ′ ( t +1) V ∞ N ( t +1 , ℓ ′ ) (1.16)= 1 k X ( t +1) V ∞ N ( t +1) k × ω k,ℓ ( t +1) V ∞ N ( t +1 , ℓ ) V ∞ N ( t, k ) (1.17) for t ≥ , k, ℓ ∈ { , . . . N } . (2) Let ω ∈ Ω . For the chain with transition (1.16) starting at time s with law ν N ( s, · ) , wehave for t ≥ s , P ω ( j t = ℓ ) = ν N ( t, ℓ ) , ℓ = 1 , . . . N. This is proved in Section 2.3.Note that, if Z N (0 , · ; t, ⋆ ) denotes the vector ( Z N (0 , t, ⋆ ) , · · · , Z N (0 , N ; t, ⋆ )) ∗ , then Z N (0 , · ; t ; ⋆ ) = Π( t ) , (1.18)where Π( t ) is defined in (1.4). This representation allows us to define a polymer measure withmore general initial conditions. The above theorem can be easily adapted to this setting. Remark 1.4.
The law m N = m N, is the long-time limit of the endpoint distribution, whichappears as a fixed point of a transfer operator on Z d for the nearest neighbor graph in [4] , and forlong range graphs in [3] . (Of course, for a finite state space, there is no question of localizationand the disorder is always strong.) The previous theorems provide much more information inthis simplified framework. ANDOM POLYMERS ON THE COMPLETE GRAPH 6
Case of α -Stable Environments. We consider now the particular cases when the en-vironment follows a stable law of index α ∈ (0 , S α is supported on R + and can be defined via its Laplace transform: if S is distributed according to S α , then E e − λS = e − λ α , (1.19)for all λ ≥
0. In particular, if S , · · · , S N are N independent S α -distributed random variables,then N − /α N X i =1 S i law = S α , (1.20)and, more generally, N X i =1 a i S i law = S α , (1.21)provided P Ni =1 a αi = 1 and a i ≥
0. The tail of S α is known to decay polynomially, P [ S > x ] ∼ − α ) x − α , as x → ∞ . Furthermore, S α is the limit of properly normalized sums of i.i.d. random variablesexhibiting similar decay.It turns out that this choice of environment makes the model solvable, in the sense that thelaw of the (properly normalized) partition function is explicit. The decomposition (1.7) has tobe slightly modified to reveal the rich structure of this version of the model: let S N ( t, j ) := Z N ( t, j ) || Z N ( t − || α , φ N ( t ) := log || Z N ( t ) || α , (1.22)so that log Z N ( t, j ) = log S N ( t, j ) + φ N ( t − . (1.23)There are many reasonable manners to measure the mean height of the polymer. However the α -norm yields an unexpectedly simple description. The full probabilistic structure of the stablecase is detailed in the next theorem. Theorem 1.5.
Suppose { ω i,j ( t ) : t ≥ , ≤ i, j ≤ N } is an i.i.d. family of S α -distributedrandom variables. Then, (1) { S N ( t, j ) : t ≥ , ≤ j ≤ N } is an i.i.d. family of S α -distributed random variables.Moreover, the terms in the sum (1.23) are independent. (2) Starting from any state, the Markov chain X N,α ( · ) reaches equilibrium instantaneously.In fact, ( X N,α ( t )) t ≥ is an i.i.d. sequence in ¯ B α for all starting point X N,α (0) . (3) { φ N ( t ) : t ≥ } is a random walk with i.i.d jumps { Υ N ( t ) : t ≥ } distributed as Υ N law = log k S N k α , (1.24) where S N is a N -vector with i.i.d. S α -distributed coordinates. (4) v N = E [Υ N ] , σ N = V ar [Υ N ] . (5) The invariant law m N,α has the same distribution as S N || S N || α . (6) The sequence { ( V ∞ N,α ( t, j )) Nj =1 } t ≥ is i.i.d. with common distribution m N,α . This is proved in Section 3.1.Note that, from the product of random matrices point of view, our results look very close to[14] in spirit, although only the symmetric stable case is treated there. However, an inspectionof their proofs shows that they do not cover the case of positive totally asymmetric stable lawsstudied here.As the velocity and variance from Theorem 1.2 are now explicit, we can try to obtain theirasymptotics when N grows. ANDOM POLYMERS ON THE COMPLETE GRAPH 7
Proposition 1.6.
Assume { ω i,j ( t ) : t ≥ , ≤ i, j ≤ N } is an i.i.d. family of S α -distributedrandom variables. Let c α := Γ( α ) sin παπα . (1.25) Then, as N → ∞ , v N = α − (cid:0) log N + log log N + log c α (cid:1) + o (1) , (1.26) σ N = π α log N + o ( 1log N ) . This is proved in Section 3.2.We now state the convergence of the rescaled random walk or polymer height to a L´evy process:
Theorem 1.7.
Assume { ω i,j ( t ) : t ≥ , ≤ i, j ≤ N } is an i.i.d. family of S α -distributedrandom variables. Then, for any sequence k N → ∞ , we have that φ N ( k N τ ) − γ N k N τk N / log N α → S ( τ ) , (1.27) in law in the Skorohod topology, where φ N is defined in (1.22) and γ N = 1 α log (cid:18) N log N Γ(1 − α ) (cid:19) + log k N α log N and S ( · ) is a totally asymmetric L´evy process with exponent ψ ( u ) = Z ∞ ( e iux − dxx + Z ( e iux − − iux ) dxx . (1.28)The proof is very close to the one of the corresponding statement in [16] and is given in Section3.3.Finally, we obtain a Poisson-type convergence result for the invariant measure m N,α in the caseof an S α -distributed environment. This is presented in Section 3.4.1.4. Perturbative results.
We will now study the case of environments that are perturbationsof the S α laws.Let α ∈ (0 ,
1) and suppose { ω ij ( t ) : t ≥ , ≤ i, j ≤ N } is an i.i.d. family of random variableswith a common Laplace transform ϕ ( u ) = E (cid:2) exp (cid:8) − u ω i,j ( t ) (cid:9)(cid:3) , u ≥ , such that 1 − ϕ ( u ) ∼ u α , u → + . (1.29)We view such an environment as a perturbation of the α -stable distributed environment, since ω lies in the domain of attraction of the α -stable law. It is important to note that E ω i,j ( t ) = ∞ ,therefore the various partition functions are not integrable and cannot be normalized. Inparticular, the models we consider are outside the range of application of the main techniquesin the field of directed polymers [15]. Let u α denote the distribution function of the logarithmof an S α random variable u α ( x ) = P ( S α > e x ) , x ∈ R , (1.30)and let U N denote the front profile of the polymer, U N ( t, x ) := 1 N N X j =1 log Z N ( t,j ) >x , t ∈ N , x ∈ R . (1.31)The random function x U N ( t, x ) is the (inverse) distribution function of the polymer heightfunction. Theorem 1.8.
Suppose the ω ’s satisfy (1.29) for a given α ∈ (0 , . ANDOM POLYMERS ON THE COMPLETE GRAPH 8 (1)
Then, for all t ≥ and any i ∈ { , · · · , N } , we have Z N ( t, i ) || Z N ( t − || α law −→ S α . Moreover, for each k ≥ and any sequence K N ⊂ { , · · · , N } with | K N | = k , we have (cid:26) Z N ( t, i ) || Z N ( t − || α : i ∈ K N (cid:27) law −→ S ⊗ kα . (2) Furthermore, for all t ≥ , we have, with φ N from (1.22) and u α from (1.30) U N (cid:0) t, x + φ N ( t − (cid:1) → u α ( x ) a.s., as N → ∞ , uniformly in x . Comment : Since the polymer is pulled by the largest values, the height function is expected togrow like a moving front. Roughly, the front behaves like a wave traveling at speed v N , and onecould try to look at it at time t around location v N t . But if the cardinality N of the monomerstate space is large, the actual front can be rather different from v N t , so it is natural to lookat it relative to a suitable random location φ N ( t − ω close to an α -stable law (i.e., an exactly solvable model), the polymer height function seenfrom this location φ N ( t −
1) converges – without scaling – to the exactly solvable model.Theorem 1.8 is proven in Section 4.2. In Section 4.1 we will also prove more complete resultsin the case of stable environments, including a fluctuation theorem; see Proposition 4.1.2.
General environments with fixed N We note once for all that Ψ α is a continuous bijection from ¯ B ≡ ¯ B to ¯ B α for any α > α = 1. The objects constructed on ¯ B can then be projectedon the other α -symplexes yielding the results for general values of α . In particular, the relation V ∞ N,α = Ψ α ( V ∞ N,β ) appearing in part 2 of Theorem 1.1, follows immediately. Hence, we willrestrict to α = 1 for the rest of this section.We follow the well-known theory of product of random matrices, see e.g. [28, 29].2.1. Asymptotics of the Markov chain and stochastic contractivity.
In this sectionwe prove Theorem 1.1, using the following lemma, cf. section I in [28], which is the involvedstep. We reproduce the proof in the appendix as it contains some crucial ideas, in particular acontraction property. (And, moreover, it is beautiful.)
Lemma 2.1.
There exists a r.v. V ∞ N taking values in B such that, for all v ∈ ¯ B , Π( t ) · v converges a.s. to V ∞ N as t → ∞ . The convergence is a.s. uniform in v ∈ ¯ B . This immediately implies the almost sure convergence stated in part 2 . of Theorem 1.1. Notethat we can conclude that X N ( t ) converges in law to m N as t → ∞ . Lemma 2.2.
The law m N of V ∞ N is the unique invariant probability, i.e., the unique probabilitymeasure on ¯ B such that, for all bounded continuous f : ¯ B → R , Z ¯ B E (cid:2) f (cid:0) X · v (cid:1)(cid:3) dm N ( v ) = Z ¯ B f ( v ) dm N ( v ) . The law m N is usually called the Furstenberg measure. Proof.
From Lemma 2.1, V ′ = lim t Π(2 , t ) · x converges a.s., and has the same law m N . Theequality X (1) · V ′ = V ∞ N is the claimed invariance property. Moreover, if m ′ N is another invariantlaw, we get by iterating t times, Z ¯ B E (cid:2) f (cid:0) Π( t ) · x (cid:1)(cid:3) dm ′ N ( x ) = Z ¯ B f ( x ) dm ′ N ( x ) . ANDOM POLYMERS ON THE COMPLETE GRAPH 9
By dominated convergence, the left-hand side converges to E f ( V ∞ N ), and we can conclude that m ′ is the law of µ ∞ . Lemma 2.3.
The law m N is invariant for the Markov chain ( X N ( t ); t ≥ , and the chain X N with initial law m N is ergodic. For any bounded continuous f : ¯ B → R and any initialcondition X N (0) ∈ ¯ B , lim t →∞ t t X s =1 f (cid:0) X N ( s ) (cid:1) = E (cid:2) f ( V ∞ N ) (cid:3) , P − a . s . (2.1) Proof.
We can write (1.10) as X (0) α · V ∞ N,α = V ∞ N,α ◦ θ − . Transposing this identity and usingthat ( X (0) , V ∞ N , V ∞ N ◦ θ − ) law = ( X (1) , ←− V ∞ N , ←− V ∞ N ◦ θ ), we see that ←− V ∞ N · X (1) = ←− V ∞ N ◦ θ . Since ←− V ∞ N and X (1) are independent and ←− V ∞ N is stationary with law m N , this equality impliesthat m N is invariant.Ergodicity is shown in Lemma 3.3 in [28]. Finally, the pointwise ergodic theorem (2.1) followsalso from the previous and the contraction property, see Appendix A.2.2.2. Free energy and Lyapunov exponents.
The Perron-Frobenius eigenvalue of the (strictly)positive matrix Π( t ) is the r.v. λ PF N ( t ) = min x ∈ ( R ∗ + ) N max ≤ i ≤ N (cid:0) Π( t ) x (cid:1) i x i = max x ∈ ( R ∗ + ) N min ≤ i ≤ N (cid:0) Π( t ) x (cid:1) i x i . We start by stating that all coefficients of the matrix Π( t ) grow like the Perron-Frobeniuseigenvalue. Lemma 2.4. [29, Lemma 2.1]
Fix N . We have for all t ≥ , ≤ log max i,j ≤ N Π i.j ( t )min i,j ≤ N Π i.j ( t ) ≤ log max i,j ≤ N ω i.j (1)min i,j ≤ N ω i.j (1) + log max i,j ≤ N ω i.j ( t )min i,j ≤ N ω i.j ( t ) , (2.2) and for all y ∈ ( R ∗ + ) N , ≤ max i ≤ N (Π( t ) y ) i min i ≤ N (Π( t ) y ) i ≤ max i,j ≤ N ω i.j (1)min i,j ≤ N ω i.j (1) . (2.3) Moreover, sup t ≥ (cid:12)(cid:12) log λ PF N ( t ) − log k Π( t ) k (cid:12)(cid:12) < ∞ a . s . (2.4) Proof.
By positivity, we see that for all m, m ′ , n, n ′ ≤ N ,Π m,n ( t ) = X k,ℓ ≤ N ω m,k (1)Π k,ℓ (1 , t − ω ℓ,n ( t ) ≤ max i,j ≤ N ω i.j (1)min i,j ≤ N ω i.j (1) × Π m ′ ,n ( t )Π m,n ( t ) ≤ max i,j ≤ N ω i.j ( t )min i,j ≤ N ω i.j ( t ) × Π m,n ′ ( t ) . This implies (2.2)–(2.3). By Perron-Frobenius theorem, ∃ x > k x k = 1 and x ∗ Π( t ) = λ PF N ( t ) x ∗ . Then, on the one hand, x ∗ Π( t ) = λ PF N ( t ) x ∗ = λ PF N ( t ) , while we can estimate x ∗ Π( t ) = X i ≤ N x i (Π( t ) ) i ≤ k Π( t ) k ∞ ANDOM POLYMERS ON THE COMPLETE GRAPH 10 and x ∗ Π( t ) = X i ≤ N x i (Π( t ) ) i ≥ min i (Π( t ) ) i k x k = min i (Π( t ) ) i (2.3) ≥ min i,j ≤ N ω i.j (1)max i,j ≤ N ω i.j (1) × max i (Π( t ) ) i Then the claim follows using that N − | y | ≤ | y | ∞ ≤ | y | with y = Π( t ) ∈ R N .We are now able to show the existence of the free energy and the Gaussian fluctuations of thelogarithm of the partition function. Proof of Theorem 1.2.
For a N × N -matrix χ , set k χ k = P Ni,j =1 | χ i,j | . Since the norm k · k is submultiplicative, we see that the doubly indexed sequence log k Π( s, t ) k (0 ≤ s ≤ t ) issubbadditive and, by the subbadditive ergodic theorem (see e.g. the nice proof in [38]), it followsthat t − log k Π( t ) k converges a.s. to a limit v N , and similarly for the entries t − log k Π i,j ( t ) k by Lemma 2.4. Moreover, from Theorem 3 in [28], a central limit theorem holds. It then sufficesto recall that Z N ( t, j ) = (Π( t ) ∗ ) j .Strict positivity of the variance follows from two results in [28]. By Corollary 3 therein, σ N = 0implies that ( e − tv N k Π( t ) k ) t ≥ is a tight sequence in (0 , ∞ ). By Theorem 5, this is equivalentto a certain geometric property of the support of the law of X , which is clearly not satisfied for X with non-constant, i.i.d. entries.It remains to prove (1.11). By Lemma 2.4, we have v N = lim t →∞ t log k Z N ( t ) k = lim t →∞ t t X s =1 log k Z N ( s ) k k Z N ( s − k . ) = lim t →∞ t t X s =1 log k X ( s ) ∗ X N ( s − k . = E log k X (1) ∗ X N (0) k , which is equal to the RHS of (1.11).2.3. Infinite volume measure.
In this section we prove the existence of the infinite volumepolymer measure and of a co-variant measure.
Proof of Theorem 1.3.
Recall the definition (1.12) of the finite horizon P2L polymer measure.It is well known, and easily checked, that P ω ,i ; T,⋆ is a time-inhomogeneous Markov chain on { , . . . , N } , with 1-step transitions given for 0 ≤ t < T by P ω ,i ; T,⋆ ( j t +1 = ℓ (cid:12)(cid:12) j t = k ) = ω k,ℓ ( t + 1) Z N ( t + 1 , ℓ ; T, ⋆ ) P ≤ m ≤ N ω k,m ( t + 1) Z N ( t + 1 , m ; T, ⋆ )= ω k,ℓ ( t + 1) Z N ( t +1 ,ℓ ; T,⋆ ) P ℓ ′ Z N ( t +1 ,ℓ ′ ; T,⋆ ) P ≤ m ≤ N ω k,m ( t + 1) Z N ( t +1 ,m ; T,⋆ ) P ℓ ′ Z N ( t +1 ,ℓ ′ ; T,⋆ ) But, a.s., (cid:18) Z N ( t + 1 , ℓ ; T, ⋆ ) P ℓ ′ Z N ( t + 1 , ℓ ′ ; T, ⋆ ) (cid:19) Nℓ =1 = Π( t + 1 , T ) · −→ V ∞ N ( t + 1) ANDOM POLYMERS ON THE COMPLETE GRAPH 11 as T → ∞ , so the above transition converges, P ω ,i ; T,⋆ ( j t +1 = ℓ (cid:12)(cid:12) j t = k ) −→ ω k,ℓ ( t + 1) V ∞ N ( t + 1 , ℓ ) P Nℓ ′ =1 ω k,ℓ ′ ( t + 1) V ∞ N ( t + 1 , ℓ ′ ) . This proves that the finite horizon P2L polymer measure converges to the Markov chain P ω given by the transition probabilities (1.16). In order to obtain (1.17), one can use a shiftedversion of (1.10), X ( t + 1) · V ∞ N ( t + 1) = V ∞ N ( t ) , to rewrite the denominator in the RHS of (1.16) as N X ℓ ′ =1 ω k,ℓ ′ ( t + 1) V ∞ N ( t + 1 , ℓ ′ ) = V ∞ N ( t, k ) k X ( t + 1) V ∞ N ( t + 1) k . We end by proving the second part of Theorem 1.3. Note that (1.15) writes ν N ( t, j ) := ←− V ∞ N ( t, j ) V ∞ N ( t, j ) ←− V ∞ N ( t ) ∗ V ∞ N ( t ) , (2.5)where ←− V ∞ N ( t ) ∗ V ∞ N ( t ) = P Nk =1 ←− V ∞ N ( t, k ) V ∞ N ( t, k ), so that, with (1.17), ν N ( t, k ) P ω ( j t +1 = ℓ (cid:12)(cid:12) j t = k ) = ←− V ∞ N ( t, k ) × ω k,ℓ ( t +1) × V ∞ N ( t +1 , ℓ ) k X ( t +1) V ∞ N ( t +1) k × (cid:0) ←− V ∞ N ( t ) ∗ V ∞ N ( t ) (cid:1) . (2.6)Summing over k and using (1.13) we get N X k =1 ν N ( t, k ) P ω ( j t +1 = ℓ (cid:12)(cid:12) j t = k ) = ←− V ∞ N ( t +1 , ℓ ) × k X ( t +1) ∗ ←− V ∞ N ( t ) k × V ∞ N ( t +1 , ℓ ) k X ( t +1) V ∞ N ( t +1) k × (cid:0) ←− V ∞ N ( t ) ∗ V ∞ N ( t ) (cid:1) (2.7)Summing now over ℓ we derive the identity k X ( t +1) V ∞ N ( t +1) k (cid:0) ←− V ∞ N ( t ) ∗ V ∞ N ( t ) (cid:1) = k X ( t +1) ←− V ∞ N ( t ) k (cid:0) ←− V ∞ N ( t +1) ∗ V ∞ N ( t +1) (cid:1) (2.8)Using (2.8) back in the RHS of (2.7) we see that it is equal to ν N ( t + 1 , ℓ ), proving the claim. Remark 2.1.
We have in fact a time-reversal property. Define the (time-inhomogeneous)transition probability ←− P ω on { , , . . . N } by ←− P ω ( j t +1 = k (cid:12)(cid:12) j t = ℓ ) = ω k,ℓ ( t + 1) ←− V ∞ N ( t, k ) ←− V ∞ N ( t + 1 , ℓ ) k X ( t +1) ∗ ←− V ∞ N ( t +1 k Then, by (2.8) , the equality (2.6) writes ν N ( t, k ) P ω ( j t +1 = ℓ (cid:12)(cid:12) j t = k ) = ν N ( t + 1 , ℓ ) ←− P ω ( j t +1 = k (cid:12)(cid:12) j t = ℓ ) , from which stationarity follows immediately. The time-reversed of the chain discussed in item2) of Theorem 1.3 is the chain with transitions ←− P ω and starting at time from the law ν N (0 , · ) . Exact solution for stable laws
We consider the particular cases when ω i,j ( t ) law = S α , the stable law of index α ∈ (0 , R + and that, for λ > E [ e − λ S α ] = e − λ α . (3.1) ANDOM POLYMERS ON THE COMPLETE GRAPH 12
The random walk representation.
This section is devoted to the proof of Theorem 1.5which summarizes the probabilistic structure of the model.Let by F t be the σ -field generated by the ω i,j ( s ) for s ≤ t and all i, j . Property (1.21) directlyimplies that for each j , conditionally on F t , S N ( t + 1 , j ) has law S α , for all 1 ≤ j ≤ N . To dealwith the N -vector, we fix λ j > ≤ j ≤ N and we compute, E [ e − P Nj =1 λ j Z N ( t +1 ,j ) |F t ] = N Y i,j =1 E [ e − λ j Z N ( t,i ) ω i,j ( t +1) |F t ]= exp {− N X j =1 λ αj N X i =1 Z N ( t, i ) α } (by (3.1)) . (3.2)Then E [ e − P Nj =1 λ j S N ( t +1 ,j ) |F t ] = exp {− N X j − λ αj } , (3.3)which shows that, conditionally on F t , S N ( t + 1 , j ) with 1 ≤ j ≤ N ) have law S α for all1 ≤ j ≤ N and are conditionally independent. As a consequence, the random variables { S N ( t + 1 , j ) : t ≥ , j = 1 , · · · , N } are i.i.d. with common law S α . This proves Proposition1.5, part 1.Now, recall the identity (1.23)log Z N ( t, j ) = log S N ( t, j ) + φ N ( t − , (3.4)where φ N ( t ) denotes the height of the polymer at time t : φ N ( t ) = log k Z N ( t ) k α . (3.5)By definition of the α -norm, φ N ( t + 1) = α − log( N X i =1 Z ( t + 1 , i ) α )= α − log N X i =1 Z ( t + 1 , i ) α k Z N ( t ) k αα + log k Z N ( t ) k α = α − log N X i =1 S N ( t + 1 , i ) α + φ N ( t ) (3.6)= log k S N ( t + 1) k α + φ N ( t ) . (3.7)where S N ( t ) denotes the vector ( S N ( t, , · · · , S N ( t, N )). From the independence observedabove, the sequence { Υ N ( t ) : t ≥ } defined byΥ N ( t ) = log k S N ( t ) k α is i.i.d., and ( φ N ( t )) t is a random walk with jumps Υ N ( · ). The identity (3.4) shows that { log Z j ( t ) : j = 1 , · · · , N } is an independent N -sample of the α -stable law with an independentshift by φ N ( t − S = ( S , · · · , S N ) be a vectorwith i.i.d. entries following the law S α and let X = S || S || . Then, X (0) · X = X (0) · S = X (0) S || X (0) S || = ˜ S || ˜ S || , ANDOM POLYMERS ON THE COMPLETE GRAPH 13 where ˜ S = X (0) S || X (0) S || α . By the conditioning argument above, we see that ˜ S has the same distri-bution as S . Hence, X and X (0) · X have the same distribution and the law of X is indeed theunique invariant measure for the system.3.2. Asymptotic of the Lyapunov exponent for large N : We will prove Proposition 1.6,namely, αv N = log N + log log N + log c α + o (1) , α σ N = π N + o ( 1log N ) , with c α from (1.25). Recall from Theorem 1.5 that v N = E Υ N = E log k S N k α = α − E log N X j =1 S α ( j ) , for S N = ( S ( j ); j = 1 , · · · , N ) an N -sample of S α independent random variables, and σ N = E (cid:2) (Υ N − v N ) (cid:3) (3.8)Note that P [ S α > x ] ∼ x Γ(1 − α ) so that the S ( j ) α ’s are in the domain of attraction of the totallyasymetric stable law of index 1 that we will denote by S . This is the stable law of index α = 1,with characteristic function given for u ∈ R by E e iu S = exp (cid:26)Z ∞ ( e iux − dxx + Z ( e iux − − iux ) dxx (cid:27) (3.9)= exp (cid:26) iCu − π | u | (cid:8) i π sign( u ) ln | u | (cid:9)(cid:27) for some real constant C defined by the above equality. It takes real values, not only positiveones. We can then appeal to a general fluctuation result: Proposition 3.1.
Suppose ( X i ) i is a family of i.i.d. positive random variables such that P [ X >x ] ∼ x − L ( x ) , with L ( x ) a slowly varying function. Let a N = inf { x : P [ X > x ] ≤ /N } and b N = N E [X X < a n ] . (3.10) Then, P Ni =1 X i − b N a N law −→ S . (3.11) Proof.
This is a particular case of [21, Th. 3.7.2].In our case, we can replace a N , b N by their leading order (denoting them with the same symbolfor simplicity): with a N = N Γ(1 − α ) and b N = N log N Γ(1 − α ) , we have S N := Γ(1 − α ) N × ( N X j =1 S ( j ) α − N log N Γ(1 − α ) ) law −→ S , (3.12)as N → ∞ . In particular, P Nj =1 S ( j ) α N log N → − α ) , (3.13)in probability, as N → ∞ . The asymptotics for v N follows from next proposition: Lemma 3.1. lim n →∞ E " log P Nj =1 S ( j ) α N log N = − log Γ(1 − α ) . (3.14) ANDOM POLYMERS ON THE COMPLETE GRAPH 14
Proof.
To derive convergence of moments from the convergence in probability above, we proveuniform integrability, following arguments of [14, proof of Prop. 2.8]. It suffices to show thatsup N E (cid:0) Σ N / ( N log N ) (cid:1) a < ∞ (3.15)both for some for a > a <
0, where Σ N = P Ni =1 S ( i ) α .For the first purpose, let ˆΣ N = P Ni =1 S αi { S αi ≤ Nu } and note that P (Σ N = ˆΣ N ) ≤ N P ( S αi > N u ) ∼ Γ( α ) sin( πα ) πu as N → ∞ . Use Markov inequality and bound, for u > P (Σ N / ( N log N ) > u ) ≤ u − E (cid:0) ˆΣ N / ( N log N ) (cid:1) + P (Σ N = ˆΣ N )= O ( u − log u ) , which implies (3.15) for a ∈ (0 , a = −
1: write by Fubini’s theorem E h(cid:0) Σ N / ( N log N ) (cid:1) − i = Z ∞ E e − t Σ N / ( N log N ) dt = Z ∞ g (cid:0) t/ ( N log N ) (cid:1) N dt, where g ( t ) = E exp {− t ( S α ) α } ≤ − C t | log t | t ∈ (0 , / ,C t ∈ [1 / , log N ] ,t − γ , t > log N, (3.16)for some C > , C ∈ (0 ,
1) and γ >
0. The bound for t ∈ (0 , /
2) follows from the explicit rateof decay of the tails of S α , whereas the one for t > S α at s is O ( s ). Plugging the bounds (3.16) into the above integral, we get (3.15) for a = − αv N ∼ log N + log log N − log Γ(1 − α ).As for the variance, first note that, by [14, Lemma 2.6] (or even the arguments in the proofabove), log P Nj =1 S ( j ) α has finite variance. Now, with Σ N as in the proof of the previous lemma,log (cid:18) Σ N c α N log N (cid:19) = log (cid:18) N S N (cid:19) = F N ( S N ) , (3.17)where S N is defined in (3.12), c α = 1 / Γ(1 − α ) and F N ( x ) = log(1 + x log N ). Let L > N × F N ( x ) are uniformly bounded and uniformly continuous on { x ≤ L √ log N } . Assume that, on our probability space, S N → S almost surely. Then,log N E (cid:2) F N ( S N ) | S N |≤ L √ log N (cid:3) ∼ log N E (cid:2) F N ( S ) | S N |≤ L √ log N (cid:3) (3.18) ∼ log N Z ∞ F N ( y ) dyy . (3.19)By a simple change of variable, the last quantity equals R ∞ log (1 + y ) dyy = π .3.3. Scaling of the polymer height.
Proof.
Theorem 1.7. In this section, we proceed as in [16], Theorem 3.2.We first expand Υ N from (1.24). Recall thatΥ N = 1 α log N X j =1 S αj , ANDOM POLYMERS ON THE COMPLETE GRAPH 15 for S , · · · , S N independent S α -distributed random variables. From (3.12), we get N X j =1 S αj = N log N Γ(1 − α ) (cid:26) S N log N (cid:27) with S N converging to S , the totally asymmetric stable law of exponent 1 given by (3.9) .Hence, Υ N = 1 α log (cid:18) N log N Γ(1 − α ) (cid:19) + S N α log N + o (1 / log N )so that, for all t ≥ S N ( t ) := α log N { Υ N ( t ) − log β N } → S , (3.20)with β N = (cid:16) N log N Γ(1 − α ) (cid:17) /α . This gives the scaling limit of the jumps of the walk φ N . To get thescaling limit for the walk itself, we use basic convergence theorems of independent incrementsprocesses. We apply Theorems 3.2 in [31], taking Y nm , γ n ( t ) in formula (3.1) in that paper equal S N ( m ) , k N t respectively. We derive the desired convergence (1.27) after a few manipulations.3.4. Asymptotics of the invariant measure.
Recall the definition X N,α ( t ) = Z N ( t ) || Z N ( t ) || α = Z N ( t ) (cid:16)P j Z N ( t, j ) α (cid:17) /α . Letting S N ( t, j ) := Z N ( t,j ) || Z N ( t − || α and S N ( t ) = ( S N ( t, , · · · , S N ( t, N )), we get X N,α ( t ) = S N ( t ) (cid:16)P j S N ( t, j ) α (cid:17) /α , where we recall that { S N ( t, j ) : t ≥ , ≤ j ≤ N } is an i.i.d. family of S α -distributed randomvariables.We borrow a (basic) version of Theorem 5.7.1 from [34]: Lemma 3.2.
Let ( X i ) i be i.i.d. random variables with common distribution function F and let u n = u n ( τ ) be such that n [1 − F ( u n ( τ ))] → τ. (3.21) Then, the point process n X i =1 δ u − n ( X i ) (3.22) converges weakly to the Poisson point process (PPP) on R + with intensity measure the Lebesguemeasure. In our setting, 1 − F ( u ) ∼ cu − α as u → + ∞ with c = Γ(1 − α ) − so that we can choose u n ( τ ) = c /α n /α τ − /α , u − n ( x ) = cnx − α . (3.23)Let ( S i ) i be an i.i.d. family of S α -distributed random variables. Then, according to the aboveresult, the point process P N := N X i =1 δ cNS − αi (3.24)converges weakly to a Poisson point process ( τ i ) i with intensity dτ on R + . ANDOM POLYMERS ON THE COMPLETE GRAPH 16
A more familiar formulation is that { S i N /α } converges to a PPP with intensity proportional to s − (1+ α ) . Accordingly, it is tempting to rewrite X ∞ N,α ( t ) = N − /α S N ( t ) (cid:16)P Nj =1 ( N − /α S N ( t, j )) α (cid:17) /α , (3.25)but the denominator diverges, as P Nj =1 S N ( t, j ) α N log N → − α ) , (3.26)so that it is not unreasonable to expect(log N ) /α × X N,α ( t ) = N − /α S N ( t ) (cid:18) P Nj =1 S N ( t,j ) α N log N (cid:19) /α (3.27)to converge in some sense. Since the limit would have infinitely many points in the neighborhoodof 0, we perform a non-linear transformation on the point process, and consider σ i = − log τ i . Proposition 3.2.
Assume ω i,j ∼ S α . We have the convergence of the point process N X i =1 δ α log X ∞ N,α ( i )+log N law −→ X i ≥ δ σ i (3.28) with ( σ i ) i a Poisson point process with intensity e − σ dσ on R . ( log = log log .) We make a comment on localization: Directed polymers on the lattice in strong disorder regimehave macroscopic atoms, in the sense that the favorite sites at the ending time have massbounded away from 0 [17, 4, 3]. Here, the largest mass vanishes at order (log N ) − /α . Local-ization is thus rather weak in our model.4. Perturbative results
As a preliminary, we show an additional result when the environment is distributed as thestable law.4.1.
Wave front in the α -stable case. Recall U N ( t, x ) , u α and φ N from (1.31), (1.30) and(1.22) respectively. Proposition 4.1.
Assume ω i,j ∼ S α , and fix t ≥ arbitrary. (1) Conditionally on F t , we have a.s., as N → ∞ , U N (cid:0) t, x + φ N ( t − (cid:1) → u α ( x ) , uniformly in x . (2) As N → ∞ , log N × h U N (cid:0) t, x + ( t − v N + φ N (0) (cid:1) − u α ( x ) i law −→ u ′ α ( x ) Z . where Z is distributed as a sum of t − independent S random variables (see definition (3.9) ) i.e., equal in law to ( t − S up to a shift.Proof. Let X N ( t, j ) (3.4) ≡ log S N ( t, j ) := log Z N ( t, j ) − φ N ( t − { X N ( t, j ) : t ≥ , ≤ j ≤ N } are i.i.d.with common law log S α . Hence, conditionally on F t − , U N ( t, x + φ N ( t − u α ( x ). Pointwise convergence in item(1) then follows from the law of large numbers. Further, pointwise convergence of monotone ANDOM POLYMERS ON THE COMPLETE GRAPH 17 functions to a continuous limit is uniform on compact by Dini’s theorem; in fact, the limit iseven uniform on R because the functions are bounded. All this yields item (1).To prove item (2), first observe that U N ( t, x + ( t − v N + φ N (0)) = N X j =1 { X N ( t,j ) >x − [ φ N ( t − − φ N (0) − ( t − v N ] } is, conditionally on F t − , a binomial r.v.. By (1.22) and (1.24), α [ φ N ( t − − φ N (0) − ( t − v N ] = α t − X l =1 [Υ N ( l ) − v N ]= t − X l =1 [log k S N ( l ) k αα − αv N ]Since the i.i.d. variables S ( l, j ) α belong to the domain of attraction of S , we can write k S N ( l ) k αα = N X j =1 S ( l, j ) α =: c α N log N + c α N S N ( l )where the variable S N ( l ) defined by this equality is such that S N ( l ) law −→ S as N → ∞ (see(3.12)). Since e αv N = c α N log N (1 + o (1)) by (1.26), we have α (cid:2) φ N ( t ) − φ N (0) − tv N (cid:3) = t X l =1 (cid:20) log (cid:18) N S N ( l ) (cid:19) + o (1) (cid:21) . From this, we deduce that α log N × [ φ N ( t − − φ N (0) − ( t − v N ] converges in law to thesum of ( t −
1) independent S random variables.Moreover, for any sequence z N →
0, the central limit theorem for binomial variables implies, N / × " N N X j =1 { αX N ( t,j ) >αx − z N } − u α ( x − z N ) law −→ N (0 , u α ( x )[1 − u α ( x )]) . Together with a suitable Taylor expansion, we see that the Gaussian fluctuations vanish at therelevant scale, log N × " N N X j =1 { αX N ( t,j ) >αx − z N } − u α ( x ) + u ′ α ( x ) · z N → . Taking z N = α [ φ N ( t − − φ N (0) − ( t − v N ] and recalling the stable limit for z N log N , wecomplete the proof of the second statement.4.2. Wave front for perturbations of the α -stable case. We give the proof of Theorem1.8. It is plain to see that hypothesis (1.29) is equivalent to ϕ ( u ) = exp {− u α (1 + ε ( u )) } with lim u → + ε ( u ) = 0 , (4.1)by considering the function ε ( u ) = − u − α log ϕ ( u ) − u . Define¯ Z N ( t, j ) := Z N ( t, j ) || Z N ( t − || α = N X i =1 a N,i ω ij ( t ) , (4.2) ANDOM POLYMERS ON THE COMPLETE GRAPH 18 where a N,i = Z N ( t − ,i ) || Z N ( t − || α . Note that that a N,i are functions of X (1) , . . . X ( t −
1) and that P Ni =1 a αN,i = 1. Now, E [exp {− u ¯ Z N ( t, j ) }|F t − ] = N Y i =1 ϕ ( ua N,i )= exp ( − N X i =1 ( ua N,i ) α (cid:0) ε ( ua N,i ) (cid:1)) (by (4.1))= exp ( − u α N X i =1 a αN,i ε ( ua N,i ) !) . In the next section we will prove the following
Lemma 4.1.
Under the above hypothesis (1.29) , sup i ≤ N {k a N,i k ∞ } ≡ sup { a N,i ; 1 ≤ i ≤ N, X (1) , . . . , X ( t − } = o (1) in probability as N → ∞ when t ≥ . With the lemma at hand, we finish the proof. Then, the computation above yields E [exp {− u ¯ Z N ( t, j ) }|F t − ] = exp (cid:8) − u α (cid:0) o (1) (cid:1)(cid:9) −→ exp {− u α } = E [exp {− u S α } ]for all u ≥
0. This implies (e.g., [23] theorem 2, p.431) that ¯ Z N ( t, j ) law −→ S α , which is the firstclaim of the theorem. Conditionally on F t − , the variables ( Z N ( t, i ) / || Z N ( t − || α : i ∈ K N )are i.i.d., so the second claim follows directly.We end by proving item (2). From the law of large numbers for triangular arrays of i.i.d.random variables, the claim follows for all real x . Then, uniformity is obtained similarly to theproof of Proposition 4.1. This ends the proof of Theorem 1.8.4.3. Proof of Lemma 4.1.
We start with a simple proof under a stronger assumption, becausethe coupling argument there makes things transparent. In a second step, we give the proof underour general assumption.4.3.1.
Coupling.
In this section we present the proof under more restrictive hypothesis: Weassume there exist constants a < b such that u α ( x − b ) ≤ P (cid:0) log ω i,j ( t ) ≤ x (cid:1) ≤ u α ( x − a ) , x ∈ R . (4.3)with u α from (1.30). This allows to couple the environment ω at time t − α -stablerandom variables s i,j on a larger probability space in such a way that c − s ij ≤ ω ij ( t − ≤ cs ij , ≤ i, j ≤ N, (4.4)with c = max { a − , b } >
1. Define˜ Z N ( t − , j ) = N X i =1 Z N ( t − , i ) s ij (4.5)Then, for some constant C , we have a N,i ≤ C ˜ Z N ( t − , i ) || ˜ Z N ( t − || α = C S i (cid:16)P Nj =1 S αj (cid:17) /α , (4.6)where the random variables S j = ˜ Z ( t − ,j ) || ˜ Z ( t − || α , j = 1 , · · · , N are now independent and S α -distributed. As noted above, the S αj ’s are in the domain of attraction of S , so that the ANDOM POLYMERS ON THE COMPLETE GRAPH 19 denominator is O (( N log N ) /α ). Recall that the sum of the S i ’s is O ( N /α ), we concludethat P Ni =1 a N,i = O (cid:16) N ) /α (cid:17) in probability. The estimate is uniform on F t − .4.3.2. Proof completed.
In this section we present the proof under the standing hypothesis(1.29).First, recall a well-known relation on tails of distribution and Laplace transform of a positiver.v. V : The equivalent hypothesis (4.1), E e − uV = exp {− u α (1 + ε ( u )) } with lim u → + ε ( u ) = 0,is itself equivalent, by the Tauberian theorem (Corollary 8.1.7 in [5]) to P [ V ≥ x ] ∼ x α Γ(1 − α ) , x → ∞ . (4.7)We note that ¯ Z N ( t, i ) defined in (4.2), satisfies (4.1) (equivalently, (1.29)). Indeed, E (exp {− u ¯ Z N ( t, j ) }|F t − ) = exp (cid:8) − u α (cid:0) N X i =1 a αN,i ε ( a N,i u ) (cid:1)(cid:9) , where we can bound the sum uniformly by max { ε ( v ); v ∈ [0 , u ] } , which vanishes as u → V = ¯ Z N ( t − , i ), which obeys the tail estimate (4.7).Thus, we can apply the theory of extreme statistics for triangular arrays of i.i.d. real variables,which implies that the maximum of ¯ Z N ( t − , i ) over i normalized by N − /α converges to aFrechet law. In complete details, we have (e.g., [9], Th. 2.28) (cid:26) ¯ Z N ( t − , i ) N /α : i ≤ N (cid:27) law −→ PPP with intensity α Γ(1 − α ) x − (1+ α ) R + . This point process has a finite right-most point and the sum of the α -powers of its termsdiverges. Hence,log a N,i ≤ max j ≤ N log ¯ Z N ( t − , j ) N /α − α log X j (cid:18) ¯ Z N ( t − , i ) N /α (cid:19) α → −∞ , N → ∞ . (4.8)This proves the lemma. Appendix A. Projective space and stochastic contractivity
For the sake of completeness, we give a proof of Lemma 2.1 and expose the principal ideasleading to it.A.1.
Contraction in the projective space.
When studying random matrix products, it isconvenient to introduce the projective action of these matrices. See, e.g., [8] for invertiblematrices, and [28] for positive matrices. Projectively, that is, when only the directions areconsidered, the elements of the positive N -dimensional orthant are represented by points of theopen and closed polygons B = { x ∈ ( R ∗ + ) N ; k x k = 1 } , ¯ B = { x ∈ ( R + ) N ; k x k = 1 } . (A.1)If g is a N × N matrix with (strictly) positive entries, we denote the projective action of g on B by the notation · , g · x = gx k gx k , (4.7) implies that the solution v N ( τ ) of N P ( ¯ Z N ( t − , j ) ≥ v N ( τ ) |F t − ) = τ is such that v N ( τ ) ∼ N Γ(1 − α ) u α uniformly over F t − . This is all what we need in the i.i.d. case to apply the theorem. ANDOM POLYMERS ON THE COMPLETE GRAPH 20 which belongs to B for all x ∈ ¯ B . The set B can be equipped with a convenient metrics d ( x, y ),which relates to the Hilbert distance (see [28, Remark below Prop. 3.1]). For x, y ∈ ¯ B let m ( x, y ) = sup { λ ≥ λy i ≤ x i , ∀ i ≤ N } ∈ [0 , , and define, with φ ( s ) = (1 − s ) / (1 + s ), d ( x, y ) = φ (cid:0) m ( x, y ) m ( y, x ) (cid:1) . Proposition A.1 (Sect. 10 in [28]) . The map d : ¯ B × ¯ B → [0 , defines a distance on ¯ B , andwe have:(i) The topology of ( B, d ) is the topology on B induced by the usual topology on R N . For x ∈ ¯ B \ B and y ∈ B , we have m ( x, y ) = 0 and then d ( x, y ) = 1 . Also, k x − y k ≤ d ( x, y ) , x, y ∈ ¯ B. Define the contraction coefficient c ( g ) of a positive N × N matrix g as c ( g ) = sup { d ( g · x, g · y ); x, y ∈ ¯ B } ∈ [0 , . Then,(ii) ∀ x, y ∈ ¯ B, d ( g · x, g · y ) ≤ c ( g ) d ( x, y ) ;(iii) If g has strictly positive entries, c ( g ) < ;(iv) For the product of two positive matrices g, g ′ , c ( gg ′ ) ≤ c ( g ) c ( g ′ ) ;(v) c ( g ∗ ) = c ( g ) . A.2.
Stochastic contractivity.
We present the proof of Lemma 2.1. Recall our principaltask consists in showing the existence of a B -valued r.v. V ∞ N such that, Π( t ) · x → V ∞ N a.s. forall x ∈ ¯ B as t → ∞ .The random sequence c (Π( t )) decreases and hence has a limit for all ω . By item (iv) ofProposition A.1, c (Π( t )) ≤ Q t − s =0 c ( X ( s )). By item (iii), each term is strictly less than 1, andby independence, we have a.s.lim sup t →∞ t − log c (Π( t )) ≤ E log c ( X ) < . (In fact, by subadditivity, the lim sup is an a.s. limit, and the value is inf t t − E log c (Π( t )).)Then, the random polygons K ( t, ω ) = { Π( t ) · x ; x ∈ ¯ B } form a decreasing sequence of compactsubsets of B , so that they have a limit, K ( ω ) = \ t ≥ K ( t ) = ∅ . If x, y ∈ K ( ω ), we have for all t , d ( x, y ) ≤ sup { d (Π( t ) · x ′ , Π( t ) · y ′ ); x ′ , y ′ ∈ ¯ B } ≤ c (Π( t )) , so the distance is 0. Finally, K ( ω ) reduces to a random point, say V ∞ N of B . In particular, V ∞ N ∈ K ( t ) implies that d (Π( t ) · x, V ∞ N ) ≤ c (Π( t )) → , so that Π( t ) · x → V ∞ N in the d -distance as well as in the Euclidean distance by item (i) ofProposition A.1. ANDOM POLYMERS ON THE COMPLETE GRAPH 21
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E-mail address : [email protected] (Gregorio Moreno, Alejandro F. Ram´ırez) Facultad de Matem´aticas, Pontificia Universidad Cat´olicade Chile, Vicu˜na Mackenna 4860, Macul, Santiago, Chile
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